Nielsen number $N(f)$ serves as a lower bound for the number of fixed points of self maps in the homotopy class of the given map $f\colon X\to X$. It is also a lower bound for the number of components of fixed point sets of all such maps. But, in general, the relative Nielsen number $N(f;X,A)$ is not a lower bound for the number of components of fixed point set of the maps in the relative homotopy class of $f\colon (X,A)\to (X,A)$.

In this paper, we introduce a new relative homotopy invariant $N^C(f;X,A)$, which is a lower bound for the number of components of fixed point set of the maps in the relative homotopy class of $f\colon (X,A)\to (X,A)$. Some properties of $N^C(f;X,A)$ will be given, which are very similar to those of $N^C(f;X,A)$.